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Math Help - Sum and products of series

  1. #1
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    Sum and products of series

    Hello,

    I was wondering if anyone could help me show the following:

    \displaystyle\sum_{i=1}^n\frac{(t+n-1)!(i-1)!}{(t+i-1)!}=\frac{(n+t-1)!-n!}{(t-1)!(t-1)},

    where t\geqslant2.

    It's an important part of a larger problem I'm working on and I'm a little stuck.

    Thanks in advance.
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by hairymclairy View Post
    Hello,

    I was wondering if anyone could help me show the following:

    \displaystyle\sum_{i=1}^n\frac{(t+n-1)!(i-1)!}{(t+i-1)!}=\frac{(n+t-1)!-n!}{(t-1)!(t-1)},

    where t\geqslant2.

    It's an important part of a larger problem I'm working on and I'm a little stuck.

    Thanks in advance.
    Induction on n.

    If we checking this formula when n=1:

    {(t+1-1)!(1-1)!}/{(t+1-1)!}=t!/t!=1

    And:

    {(1+t-1)!-1!}/{(t-1)!(t-1)}={t!-1}/{t!-1}=1

    Try to continue...
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  3. #3
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    I know it's provable by induction, however I need to derive the right hand side from the left, not just prove it if you see what I mean. Maybe the problem should be rephrased to "find a closed expression (i.e. an expression not involving the summation and product symbols) for the left hand side of that equation".
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  4. #4
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    The question asked me to derive various formulas, but when I spoke to my supervisor he said that it's apparently fine to just find a formula and then prove by induction, so thanks!
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